Infected Lymph Node or Abscess: This is the most likely diagnosis given the description of a hard, painful lump in the armpit that has been present for a week and is increasing in size. The symptoms suggest an infectious process, possibly due to a bacterial infection that has caused lymphadenitis (inflammation of the lymph nodes) or an abscess (a collection of pus).

Cat Scratch Disease: Caused by Bartonella henselae, this condition can lead to lymphadenopathy (swollen lymph nodes) that is painful and can occur in the armpit after a cat scratch or bite. The lymph nodes can become significantly enlarged and painful.
Folliculitis or Furuncle: These are skin infections involving the hair follicles, which can become inflamed and form a boil (furuncle) or a collection of boils (carbuncle). They can appear as painful, hard lumps and are common in areas with apocrine glands like the armpits.
Lipoma: Although typically painless, a lipoma (a benign tumor composed of fat tissue) could be considered if the lump is hard and has been growing. However, lipomas are usually not painful unless they press on nearby nerves or have undergone a change like necrosis.

Breast Cancer with Axillary Metastasis: Although less common, it is crucial not to miss breast cancer that has metastasized to the axillary lymph nodes. The presentation could be a hard, painless lump in the armpit, but pain can occur, especially if the tumor is large or if there is significant lymph node involvement.
Lymphoma: Both Hodgkin's and non-Hodgkin's lymphoma can present with lymphadenopathy, which may be painful. The lymph nodes can become significantly enlarged and may be hard.
Tuberculosis (TB): TB can cause lymphadenitis, which might present as a hard, painful lump in the armpit. Although less common in some regions, TB is a critical diagnosis not to miss due to its public health implications and the need for specific treatment.

Kikuchi-Fujimoto Disease: A rare, self-limiting condition that causes lymphadenitis, typically in the neck but can occur in the armpits. It's more common in young women and can present with fever and night sweats.
Kimura Disease: A rare, chronic inflammatory disorder that can cause lymphadenopathy and eosinophilia (elevated eosinophil count in the blood). It typically presents with subcutaneous nodules in the head and neck region but can occur in the armpits.
Castleman Disease: A rare disorder that involves an overgrowth of cells in the lymphatic system, which can cause lymphadenopathy. It can present with a single enlarged lymph node or with multiple lymph nodes in different parts of the body.